Equitable Coloring on Total Graph of Bigraphs and Central Graph of Cycles and Paths

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Volume 2011, Article ID 279246,5pages doi:10.1155/2011/279246

Research Article

Equitable Coloring on Total Graph of Bigraphs and

Central Graph of Cycles and Paths

J. Vernold Vivin,

1

_{K. Kaliraj,}

2

_{and M. M. Akbar Ali}

3

1_{Department of Mathematics, University College of Engineering Nagercoil, Anna University of Technology}

Tirunelveli (Nagercoil Campus), Nagercoil 629 004, Tamil Nadu, India

2_{Department of Mathematics, R.V.S College of Engineering and Technology, Coimbatore 641 402,}

Tamil Nadu, India

3_{Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore 641 062,}

Tamil Nadu, India

Correspondence should be addressed to J. Vernold Vivin,[email protected]

Received 2 December 2010; Accepted 9 February 2011

Academic Editor: Marco Squassina

Copyrightq2011 J. Vernold Vivin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The notion of equitable coloring was introduced by Meyer in 1973. In this paper we obtain interesting results regarding the equitable chromatic number χ for the total graph of

complete bigraphsTKm,n, the central graph of cyclesCCn and the central graph of paths

CPn.

1. Introduction

The central graph 1, 2 CG of a graphG is formed by adding an extra vertex on each

edge ofG, and then joining each pair of vertices of the original graph which were previously

nonadjacent.

The total graph3,4ofGhas vertex setVG∪EGand edges joining all elements

of this vertex set which are adjacent or incident inG.

If the set of vertices of a graphGcan be partitioned intok classesV1, V2, . . . , Vksuch

that eachV_i is an independent set and the condition ||V_i| − |V_j|| ≤ 1 holds for every pair

i, j, thenGis said to be equitably k-colorable. The smallest integerkfor whichGis equitable

k-colorable is known as the equitable chromatic number5–10 ofG and denoted by χG.

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2. Equitable Coloring on Total Graph of Complete Bigraphs

Theorem 2.1. Ifm≤n, the equitable chromatic number of total graph of complete bigraphsK_m,n,

χTKm,n

⎧ ⎨ ⎩

n1 ifm < n,

n2 ifmn. 2.1

Proof. LetX, Ybe the bipartition ofK_m,n, whereX{v_i: 1≤i≤m}andY {v_j : 1≤j ≤n}.

Letu_ij 1≤i≤m; 1≤j ≤nbe the edges ofv_iv_j. By the definition of total graph,TK_m,nhas

the vertex set{vi : 1≤i≤m} ∪ {v_j: 1≤j ≤n} ∪ {uij : 1≤i≤m,1≤j ≤n}and the vertices

{uij : 1≤i≤m,1≤j ≤n}inducendisjoint cliques of orderninTKm,n. Alsovi 1≤i≤m

is adjacent tov_j 1≤j ≤n.

Case 1ifmn,χTK_m,n n2. Now we partition the vertex setVTK_m,nas follows:

V1u11, u2n, u3n−1, u4n−2, . . . , un−13, un2,

V2u12, u21, u3n, u4n−1, . . . , un−14, un3,

.. .

Vnu1n, u2n−1, u3n−2, u4n−3, . . . , un−13, un1,

Vn1{v1, v2, . . . , vn},

Vn2v1, v2, . . . , vn

.

2.2

ClearlyV1, V2, . . . , Vn2are independent sets and|Vi|n1≤i≤n2satisfying the condition

||Vi| − |Vj||0, for anyi /j,χTKm,n≤n2. Since there exists a clique of ordern1 in

TKm,n.χTKm,n≥n1, also eachviofTKm,nreceives one color diﬀerent from the color

class assigned to the clique induced by{uij : 1 ≤i≤m; 1≤j ≤n}. By the definition of total

graph, eachv_i is adjacent withv_j 1 ≤ j ≤ n. Therefore,{v1, v2, . . . , v_m}and{v₁, v₂, . . . , v_n}

are independent sets and henceχTKm,n≥n2. That is,χTKm,n≥χTKm,n≥n2;

thereforeχTK_m,n≥n2. HenceχTK_m,n n2.

Case 2ifm < n,χTKm,n n1. Now we partition the vertex setVTKm,nas follows:

V1{u11, u22, u33, u44, . . . , umm} ∪vn,

V2u12, u23, u34, . . . , umm−1∪ {um1} ∪v1

,

V3u13, u24, u35, . . . , umm−2∪um−13, um2∪v2

, ..

.

Vn−1

u1n−1, u2n∪u31, u32, . . . , umm−2∪vn−2

,

Vn{u1n} ∪u21, u32, . . . , umm−1∪vn−1

,

Vn1{v1, v2, v3, . . . , vm}.

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ClearlyV1, V2, . . . , Vn1are independent sets ofTKm,n. Also|V1| |V2|· · · |Vn| m1

and|Vn1| msatisfy the condition||V_i| − |V_j|| ≤ 1, for anyi /j,χTK_m,n ≤ n1. Since

there exists a clique of order n1 inTKm,n.χTKm,n ≥ n1, that is, χTKm,n ≥

χTKm,n≥n1, thereforeχTKm,n≥n1. HenceχTKm,n n1.

3. Equitable Coloring on Central Graph of Cycles and Paths

Theorem 3.1. Ifn≥5, the equitable chromatic number of central graph of cyclesCn,

χCCn ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

n1

2 ifnis odd, n

2 ifnis even.

3.1

Proof. LetVC_n {v1, v2, . . . , v_n}andECn {e1, e2, . . . , en}be the vertices and edges ofCn

taken in the cyclic order. By the definition of central graph,CC_nhas the vertex setVC_n∪

{ui : 1 ≤ i ≤ n}, where ui is the vertex of subdivision of the edge ei and joining all the

nonadjacent vertices ofCninCCn.

Case 1nis odd. We partition the vertex setVCC_nas

V1 {v1, v2, un−2, un−1},

V2 {v3, v4, un},

V3 {v5, v6, u1, u2},

V4 {v7, v8, u3, u4},

.. .

Vn−1/2 {vn−2, vn−1, un−6, un−5},

Vn1/2 {vn, un−4, un−3}.

3.2

Clearly V1, V2, . . . Vn−1/2, Vn1/2 are independent sets of CCn. Also |V1| |V3| |V4|

· · · |Vn−1/2| 4 and |V2| |Vn1/2| 3. The inequality ||Vi| − |Vj|| ≤ 1 holds, for any

i /j, χCCn ≤ n 1/2. For each i, vi is nonadjacent with vi−1 and vi1 and hence

χCCn ≥ n1/2. That is,χCCn ≥ χCCn ≥ n1/2, χCCn ≥ n1/2.

Therefore,χCCn n1/2.

Case 2nis even. Now we partition the vertex setVCC_nas follows:

V1{v1, v2, un−3, un−2},

V2{v3, v4, un−1, un},

V3{v5, v6, u1, u2},

V4{v7, v8, u3, u4},

.. .

Vn/2{vn−1, vn, un−5, un−4}.

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ClearlyV1, V2, . . . Vn/2 are independent sets of CCn. Also|V1| |V2| |V3| |V4| · · ·

|Vn/2|4. The inequality||V_i| − |V_j|| 0 holds, for anyi /j,χCC_n≤ n/2. For eachi,v_i

is nonadjacent withvi−1andvi1and henceχCCn≥n/2. That is,χCCn≥χCCn≥

n/2,χCCn≥n/2. Therefore,χCCn n/2.

Remark 3.2. Ifn3,4, thenχCCn 2,3, respectively.

Theorem 3.3. Ifn≥5, the equitable chromatic number of central graph of pathsP_n,

χCPn

⎧ ⎪ ⎨ ⎪ ⎩

n1

2 ifnis odd, n

2 ifnis even.

3.4

Proof. LetVPn {v1, v2, . . . , vn}andEPn {e1, e2, . . . , en}be the vertices and edges ofPn.

By the definition of central graph,CPnhas the vertex setVPn∪{ui: 1≤i≤n−1}, whereui

is the vertex of subdivision of the edgee_iand joining all nonadjacent vertices ofP_ninCP_n.

Case 1nis odd. Now we partition the vertex setVCPnas follows:

V1 {v1, v2, un−2},

V2 {v3, v4, un−1},

V3 {v5, v6, u1, u2},

V4 {v7, v8, u3, u4},

.. .

Vn−1/2 {vn−1, vn−2, un−6, un−5},

Vn1/2 {vn, un−4, un−3}.

3.5

Clearly V1, V2, . . . Vn−1/2, Vn1/2 are independent sets of CPn. Also |V3| |V4| · · ·

|Vn−1/2| 4 and |V1| |V2| |Vn1/2| 3. The inequality ||Vi| − |Vj|| ≤ 1 holds, for

anyi /j,χCPn ≤ n1/2. For eachi,vi is nonadjacent withvi−1 andvi1 and hence

χCPn ≥ n1/2. That is, χCPn ≥ χCPn ≥ n1/2, χCPn ≥ n1/2.

ThereforeχCP_n n1/2.

Case 2nis even. Now we partition the vertex setVCPnas follows:

V1{v1, v2, un−3, un−2},

V2{v3, v4, un−1},

V3{v5, v6, u1, u2},

V4{v7, v8, u3, u4},

.. .

Vn/2{vn−1, vn, un−5, un−4}.

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ClearlyV1, V2, . . . Vn/2 are independent sets ofCPn. Also|V1||V3||V4|· · ·|Vn/2|4

and|V2|3. The inequalityV_i| − |V_j ≤1 holds for anyi /j,χCP_n≤n/2. For eachi,v_iis

nonadjacent withvi−1andvi1and henceχCPn≥n/2. That is,χCPn≥χCPn≥n/2,

χCPn≥n/2. Therefore,χCPn n/2.

Remark 3.4. Ifn1,2,3,4, thenχCP_n 1,2,3,3, respectively.

References

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